554-FDD-91/125

SEAS (550)/IM-91/047 (52 404)

FUTURE MISSIONS STUDIES

COMBINING SCHATTEN'S SOLAR ACTIVITY PREDICTION MODEL

WITH A CHAOTIC PREDICTION MODEL

Prepared for

GODDARD SPACE FLIGHT CENTER

By

S. Ashrafi

COMPUTER SCIENCES CORPORATION

Under

Contract NAS 5-31500

Task Assignments 51 404 and 52 404

November 1991

https://ntrs.nasa.gov/search.jsp?R=19920018978 2020-02-08T10:43:39+00:00Z

..._J"

TABLE OF CONTENTS

Section 1 - Introduction: Chaos Versus

St0chasticitv ...............

Section 2 - Solar Activity Prediction .........

Section 3 - Low Dimensional Chaos Versus

Complexity .................

Section 4 - Fractal Structure in Solar Flux

Signal ...................

Section 5 - C0mbining $chatten's Model With OurChaotic Model ...............

Glossary

References

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fq_l____ .... _N,'_N_,_'7 L_K_I_' PRECEDING PAGE BLANK NOT FILMED

LIST OF ILLUSTRATIONS

Fiqure

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5-1

5-2

5-3

5-4

5-5

Types of Noise With Their Power Spectra:

(i) White Noise, (2) Brownian Noise,

and (3) Fractional Brownian Noise ......

Tma x Increases Logarithmically With £ to

Asymptote T .................Schatten Model Versus Chaotic Model ......

Dynamo Equations (Solar Model) ........

Canonical Transformation and Lorenz

Equations ..................

The Attractor of the Turbulent Dynamo

Constructed Using Our Canonical

Transformation ...............

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5-8

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P

SECTION 1 - INTRODUCTION: CHAOS VERSUS STOCHASTICITY

Although atmospheric dynamics are governed by the same laws

of physics as planetary motion, we still forecast weather in

terms of probabilities. Because no clear relationship

exists between cause and effect in atmospheric physics,

atmospheric phenomena seem random, or stochastic. Yet,

until recently, we had little reason to doubt that, at least

in principle, weather is ultimately predictable. It was

assumed that we need only gather and process enough

information. Recently, a striking discovery changed our

perspective: Simple deterministic systems with only a few

degrees of freedom can generate random behavior.

When apparent random behavior is fundamental to the nature

of a system, such that no amount of information gathering

will make the system predictable, that system is said to be

chaotic. Perhaps paradoxically, chaos is generated by fixed

rules that do not themselves involve any elements of

chance. In principle, the future condition of a dynamic

system is completely determined by present and past

conditions. In practice, however, amplification of small

initial uncertainties makes a system with short-term

predictability unpredictable in the long term.

In the list that follows, we highlight some of the major

points in the emerging science of chaos.

• Chaos is orderly: Randomness has an underlying

geometric form.

• The discovery of chaos has created a new paradigm

in scientific modeling. On one hand, it implies new

fundamental limits on prediction. On the other hand, the

determinism inherent in chaos implies that many random and

complex phenomena are more predictable than previously

thought.

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• Chaotic theory lends order to such diverse systems

as Earth weather, the Sun, the human brain and heart, and

even the economy.

• The existence of random behavior in very simple

systems has motivated a reexamination of the sources of

randomness even in highly complex systems such as weather.

• Chaotic theory opposes the determinism of the 18th

Century philosopher-mathematician Pierre Simon de Laplace,

whose ideas greatly influenced the direction of modern

science. Laplace believed that, given the position and

velocity of every particle in the universe, one could

predict the future for the rest of time. The philosophy of

determinism holds that human behavior is predictable, given

enough data, and that free will is only an illusion.

• Twentieth Century science has seen the downfall of

Laplacian determinism in response both to the Heisenberg

uncertainty principle of quantum mechanics and to the idea

of sensitive dependence on initial conditions (discussed in

Section 2). These ideas have helped us to understand why

some apparently simple systems behave unpredictably, even

though they are subject to the same laws of motion that

allow us to predict the motion of planets, for example,

precisely. A balloon filled with air and then released is

such a system.

• The Soviet physicst Lev D. Landau was the first who

attempted to study turbulence in the 1930s. He maintained

that motion of a turbulent fluid includes many independent

oscillations. However, later research has contradicted this

idea, demonstrating random behavior even in very simple

systems, such as a coin toss.

• Early in the 20th Century, Henri Poincare

questioned the notion of determinism. Because of a growing

interest in quantum mechanics, however, the work of

J

k_

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classical physicists such as Poincare was largely ignored.Only recently have his ideas, which form the foundation of

modern chaotic theory, been seriously explored by otherscientists.

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SECTION 2 - SOLAR ACTIVITY PREDICTION

Interest in solar activity has grown in the past two decades

for many reasons. First, new evidence suggests a

correlation between solar activity and weather on Earth

(van Loon and Labitzke, 1988), although such a correlation

has not yet been convincingly established (Kerr, 1990). In

fact, we have evidence of the coincident occurrences of the

Maunder Minimum, a period of little or no solar activity

occurring from 1645 to 1715, and the "Little Ice Age," a

period of abnormally cold weather (Bray, 1971). Second,

solar activity is also studied by astronomers concerned only

with the Sun itself (Brandt, 1970). Third, and of greatest

importance to flight dynamics, solar activity changes the

atmospheric density, which has important implications for

spacecraft trajectory and lifetime prediction (Walter-

scheid, 1989).

Because of the seemingly random nature of solar activity, it

has generally been assumed that the underlying physics must

necessarily be complex as well. As a result, researchers

have turned toward statistical models to predict solar

dynamics (Withbroe, 1989, and references therein). However,

new developments in chaos and nonlinear dynamics have

demonstrated that random behavior is not always due to

complexity but rather to sensitive dependence on initial

conditions, which can sometimes cause even simple systems to

become chaotic. This view would allow us to model the

behavior of a chaotic system in terms of some invariants

directly extractable from system dynamics, without reference

to any underlying physics. Using chaos theory, we would be

able to predict short-term activity more accurately than

with statistical methods, but chaos theory puts a

fundamental limit on long-term predictions. The philosophy

behind our approach is introduced in the next section.

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SECTION 3 - LOW DIMRR$IONAL CHAOS VERSUS COMPLEXITY

In our previous communications (Ashrafi, January 1991a,

January 1991b; Ashrafi and Roszman, May 1991, June 1991a,

June 1991b, July 1991), we showed how thinking in terms of

deterministic dynamics and assuming that randomness arises

out of chaos rather than complexity lead to new approaches

to forecasting and nonlinear modeling.

Until recently, it was usually assumed that randomness was

caused by extreme complication, that is, the presence of

many irreducible degrees of freedom. This naturally led to

Kolmogorov's theory of random processes, which he defined in

terms of the joint probability distribution. The process is

deterministic if there is some value d (distribution order)

for which the probability density approaches a delta

function.

Many people speak of random processes as though they were a

fundamental source of randomness. This idea is misleading.

The theory of random processes is an empirical technique for

coping with inadequate information; it makes no statements

about the causes of randomness. As far as we know, the only

truly fundamental source of randomness is the uncertainty

principle of quantum mechanics; everything else is

deterministic, at least in principle. Nonetheless, we call

many phenomena, such as fluid turbulence, random, even

though they have no obvious connection to quantum

mechanics. It has traditionally been assumed that the

apparent randomness of these phenomena derives solely from

their complication.

We will take the practical viewpoint that randomness occurs

to the extent that a system's behavior is unpredictable,

which usually depends on the available information. With

more data or more accurate observations, a phenomenon that

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had previously seemed random might become more predictable

and, hence, less random. Therefore, we believe that

randomness is subjective. Furthermore, randomness is a

matter of degree; that is, some systems are more predictable

than others (e.g., solar activity is more predictable than

geomagnetic activity).

As originally pointed out by Poincare, many of the classic

examples of randomness are not complicated. The dynamics of

a flipping coin, for example, involve only a few degrees of

freedom. This randomness comes from sensitive dependence on

initial conditions--a small perturbation causes a much

larger effect at a later time, making prediction difficult.

When sensitive dependence on initial conditions occurs in a

sustained way, we call the result chaos. Since chaos is

defined in the context of deterministic dynamics, in some

very strict sense it is incorrect to say that chaos is

random; ultimately, uncertainty originates from something

external to the dynamics, such as measurement error or

external noise. However, sensitive dependence exaggerates

uncertainty, so that small uncertainties turn into large

ones. Because chaos amplifies noise exponentially, any

uncertainty at all is amplified to macroscopic proportions

in finite time, and short-term determinism becomes long-term

randomness: Chaos creates randomness by strongly amplifying

what we don't know.

Chaotic systems pass many classic "tests" of randomness; for

example, some simple chaotic maps produce uncorrelated time

series, with <x t xt+j> = 0 unless j = 0. Furthermore,

chaotic trajectories look random. Dissipative dynamic

systems often have the property that undisturbed

trajectories approach a subset of the state space, called an

attractor. Fluid flows, for example, have an effective

infinite dimensional state space but can have low

dimensional attractor.

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Thus, we should not distinguish chaos from randomness butshould instead distinguish systems with low dimensional

attractor from those with high dimensional attractor. With

many degrees of freedom, the statistical approach is

probably as good as any. However, if random behavior comesfrom low dimensional chaos, we can make much more accurate

forecasts than those made using statistical models.

Furthermore, the resulting chaotic models can give useful

diagnostic information about the nature of the underlying

dynamics, aiding the search for a description in terms of

first principles.

One of the leading theories proposed over the last severaldecades assumes that the Sun behaves as a hydromagnetic

dynamo (Gilman, 1985). Many dynamo-based models of varying

degrees of complexity have been proposed (Gilman, 1986).Zeldovich and Ruzmaikin (1987) have developed a low

dimensional solar dynamo model, a concept first discussed by

Ruzmaikin (1981).

Through some simple canonical transformations, we have beenable to transform dynamo equations into established Lorenz

(1963) equations. Lorenz equations are a classic example of

equations that exhibit complex chaotic behavior, includingintermittency, for a wide range of parameter values.

Additionally, Jones et al. (1985) and Weiss (1985, 1988)

have considered a model consisting of six differential

equations, as opposed to three Lorenz equations. This modelalso exhibits chaotic behavior and indicates a

period-doubling route to chaos. We had independentlyobserved period doubling through a careful study of solar

flux power spectral density, an indicator of solar activity

(Ashrafi and Roszman, May 1991).

We have determined the correlation dimension of solar flux

time series to be about 2.5, which indicates that only three

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independent variables are needed to describe its evolution.

The Lorenz and Rossler attractors have very similar

correlation dimension, also requiring three independent

variables to completely describe the system. Using a

turbulent version of the dynamo model would allow us to

model the long-time evolution of solar cycles with a low

dimensional set of ordinary differential equations.

We believe that a turbulant, or chaotic, solar dynamo model

may explain the Maunder minimas. Eddy (1976) has argued

convincingly against the idea that solar activity was

occurring but simply not observed from 1645 to 1715. He

concludes that, very likely, solar activity was actually

lacking during that period. A period of little or no solar

activity could be explained as resulting from the phenomenon

of intermittency (Ashrafi, August 1990, personal

communication), which occurs when a system alternates

between periods of laminar and chaotic behavior.

Pomeau and Manneville (1979) have established that

intermittency does indeed occur in the Lorenz system. A

detailed analysis of intermittency is given by Schuster

(1989). Therefore, the Maunder Minimum may have evolved on

the same attractor but in a region where the solar activity

as a function of time is very regular. If this view is

correct, we would expect the solar cycle to alternate

between chaotic and laminar behavior at irregular

intervals. Interestingly enough, we have evidence of other

minima occurring at certain times throughout history, for

example, the Sporer minimum, which occurred between 1460 and

1550 (Eddy, 1976). The idea of intermittency brings a

consistency to Schatten's solar dynamic model (Schatten,

1978, 1987) and our chaotic model (Ashrafi, July 1991;

unpublished manuscripts a, b, and c). We believe that no

complicated mechanism need be invoked to describe the

qualitatively diverse behavior observed in the solar cycle

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over the past 300 or, for that matter, i000 years. Thevarious behaviors observed are, in our view, simply natural

consequences of a chaotic system.

Feynman and Gabriel (1990) have recently analyzed 1500 yearsof auroral, geomagnetic, and solar activity and have proved

that solar activity does not follow quasi-periodicity,

supporting our assertion that solar activity is indeedchaotic.

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SECTION 4 - FRACTAL STRUCTURE IN SOLAR FLUX SIGNAL

Fractal geometry provides both a description and a

mathematical model of seemingly complex forms in nature.

Shapes and signals found in nature are not easily described

by traditional methods. Nevertheless, they often possess a

remarkable simplifying invariance under chanqes of

maqnification. This statistical self-similarity is the

essential quality of fractals in nature: Nature is

quantified by a fractal dimension. Fractal shapes are said

to be self-similar and independent of scale or scaling.

With the use of fractals, iteration of a very simple rule

can produce seemingly complex shapes with some highly

unusual properties. Unlike Euclidian shapes, these curves

have details on all length scales. Fractals remain by far

the best approximation of the real world. The fractal

dimension determines the relative detail or irregularity at

different scales (time or space). The addition of

irregularities on smaller and smaller scales raises the

dimension. Changes in time, however, have many of the same

similarities at different scales and changes in space.

The spectral density, S(f), gives an estimate of the mean

square fluctuations at frequency f and, consequently, of the

variations over a time scale of order I/f. Solar flux does

have fractal structure, and a direct relationship exists

between fractal dimension and logarithmic slope of the

spectral density (Figure 3-1).

The following signals are classified with respect to

randomness: (i) I/f ° white noise, the most random signal;

(2) i/f 2 Brownian noise, the most correlated of all

signals; and (3) I/f 8 fractional Brownian (FB) noise

(0.5 < 8 < 1.5), intermediate between white noise and

Brownian noise. Although its origin is, as yet, a mystery,

FB noise is the most common signal in nature.

4-110000123

s(f)

s(o

\

Figure 4-i.

log f

log f

log f

(1)

(2)

(3)

Types of Noise With Their Power Spectra:

(1) White Noise, (2) Brownian Noise, and

(3) Fractional Brownian Noise

Both i/f ° white noise and i/f 2 Brownian noise are well

understood mathematically. However, to date, no simple

mathematical models produce i/f _ FB noise. The spectral

density of solar flux has a fractal structure of dimension

2.5. This fractal structure allows us to rescale the time

and extend our prediction horizon. One might conclude that

now our extended prediction will not have as detailed a

structure as our unextended predictions. This is not

completely true, for once we have enough data to construct

the attractor in the embedding space, the extended

predictions are approximately as good as the unextended

ones. However, there would come a point Tultimate at which,

as we rescale the time, we no longer have enough data points

to construct the attractor.

4-2

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SECTION 5 - COMBINING SCHATTEN'$ MODEL WITH OUR

CHAOTIC MODEL

K. Schatten (1991) has recently developed a method for

combining his prediction model with our chaotic model. The

philosophy behind this combined model and his method of

combination is explained below.

Schatten's Model (KS). Because KS uses a dynamo to mimic

solar dynamics, accurate prediction is limited to long-term

solar behavior (i0 to 20 years).

Chaotic Model (SA). SA uses the recently developed

techniques of nonlinear dynamics to predict solar activity.

It can be used to predict activity only up to a horizon. In

theory, the chaotic prediction should be several orders of

magnitude better than statistical predictions up to that

horizon; beyond the horizon, chaotic predictions would

theoretically be just as good as statistical predictions.

Therefore, chaos theory puts a fundamental limit on

predictability.

After embedding the solar flux time series in a state space

using the Taken-Packard delay coordinate technique, one can

"learn" the induced nonlinear mapping using a local

approximation. This will allow us to make short-term

forecasting of the future behavior of our time series using

information based only on past values. The error estimate

of such a technique has already been developed by Farmer and

Sidorowich (1987).

E - C e (m + I)KT N-(m + I)/D

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where E = normalized error of prediction (0 S E _ i, wherezero is perfect prediction, and one is a predic-

tion no better than average)

m = order of local approximation

K = Kolmogorov entropy

T = forecasting window

N = number of data points

D = Dimension of the attractor

C = normalization constant

Using the Farmer-Sidorowich relation, we can find the

prediction horizon T for the zeroth order of local

approximation. Any prediction above Tma x is no better

than average constant prediction.

E(Tmax) = 1

thus,

for m = 0, K is the largest Lyapunov exponent I.

Therefore,

KT

e max N-I/D ~ 1 or T -in (N)

max KD

and

r in__n_C lmax ),D

For a finite length of data, one has to calculate the local

Lyapunov exponent. For N = 4090 point from daily solar flux

data and _ ~ 0.01 and D - 2.5 (like a Lorenz system), Tmax

- 70 days, or about 2 months. For 250 years of averaged

monthly data, Tma x - 4 years. Any prediction beyond the

indicated horizons is no better than average value. The

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connection between the local and the global Lyapunov

exponents has recently been found by Abrabanel and Kennel

(March 1991) in a form of power law as

)'(£) = )'G +

N = _£

where X(£) = local Lyapunov exponent

£ = length of observed data (observation window)

= a constant dependent to the dynamic system

(0.5 _ v _ 1.0)

c = a constant dependent to initial conditions of

the system

_G = well-known global Lyapunov exponent

= frequency of data points

Because any data are of finite length, using the

Abrabanel-Kennel power law and the Farmer-Sidorowich

relation, we can find T asmax

in (£_)T

max XG +£v

D

This means that as £ increases linearly, Tma x increases

logarithmically to a certain asymptotic T because of the

denominator C/£ v (Figure 5-1).

Therefore, our relation shows that as the asymptote T max

approaches T O , the dTmax/d£ approaches 0, and, thus,

we can find what observation window is required for

forecasting up to Tma x within some confidence level.

x 0 (6)dTmax v

dN _ 0 thus N O ~ e X 0 (6) > 2

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/

/

//

/

/

/

/

/

&

/ Io

/

/

/ I..= finite

",-3

"l'max •

T,.._ 2_

% ,

T o

8

Figure 5-1. Tma x Increases Logarithmically With £ to

Asymptote T

where X0(6) is the solution to e-x (x - i) = 6, and

_'G6 =

C_

is the scaled global Lyapunov exponent.

This result shows that any observation window greater than

£0 = N0/_ will not improve our prediction horizon

TO; so more data beyond this limit are not needed to

understand a dynamic system. This conclusion is indeed

consistent with weather prediction and also with empirical

results concluded from neural networks training.

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Combined Models: K. Schatten (1991) has introduced the

following method of combining the KS and SA models:

fpred

= avg x fA(t)SA x fKsK(t)

where fpred

KS

SA

avg

is the prediction of solar flux

is the prediction by Schatten's model

is the prediction by the chaotic model

is a constant

A(t) and K(t) are both time-dependent functions

-t

KS K(t) : 1 - e rfKS = avg

-__t

SA A(t) = e TfSA = avg

An alternate method of combining the two models is as

follows:

I. Calculate the prediction horizon for different

frequencies of data. This allows us to extend our

horizon at the expense of losing some of the data

points.

2. Predict the time series up to the specified

horizons.

3. Combine all the predictions, including the Schatten

prediction, by minimizing combined variances or by

using an appropriate Kernel density, as shown in

Figure 5-2.

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2

U-

/

(u

<uu_u..

(u _)

c_ _u,,.2 __u3 *.o

<u u. %_, u.

o_o_

EL2

oo

CL

_-- .2

0

_ _ _.2

LO

.a

,m0

0

L)

m_a

,.a

,D

I

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5-6

Dynamo Equations (Solar Model):

do3

I m = T- Mibi c -- Vo3dt

diLb b =(Mo3+Bs)_(B u +B)i

dt

Lc die = B_ib _(Be + B_)icdr

where

I"

M:

Lc:

Bb:

Moment of inertia (0:

Mutual inductance Lb:

Coil self-inductance B s"

Resistance of brush B c:

Angular velocity

Brush self-inductance

Resistance of shunt

Resistance of coil

Dynamo

. i!i_iiii!iiiiii_ii:_i_

C

i

m

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Figure 5--3 ° Dynamo Equations

5-7

(Solar Model)

Our Canonical Transformation of Dynamo Equations:

t.--_ l:t

i b --> t_y

Tco-_-Tz+-

V

Resultin_ Equations of Lorenz:

where

= (Jy - (Jx

I? = -xy+rx-y

± = xy-bz

[Bc+bs_( Lb _(_--" \--L--_c/\Bb+B s/

TMr = (r_b_+ B,v B_ V L_k___a"L'_'/\ Bc+B s/\ Bb+B s /

_v[ Lbb _ i k Bb+B s /

Figure 5-4. Canonical Transformation and Lorenz Equations

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0

/

/

\

\

0

._1

0

0

_0_._

4J

0u_u_0

0

4-_,_

0

e

I

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v

V

GLOSSARY

FB

KS

SA

fractional Brownian noise

Schatten solar prediction model

chaotic prediction model

G-I10000123

REFERENCES

Ashrafi, S., Goddard Space Flight Center, Flight Dynamics

Division, FDD-554-91-004, Future Missions Studies on

Prelimin%ry ¢omparison_ of Solar Flux Models, January 1991a

Ashrafi, S., Goddard Space Flight Center, Flight Dynamics

Division, FDD-554-91-006, Future Missions Studies on Solar

Fl_x Analysis Usina Chaos, January 1991b

Ashrafi, S., and L. Roszman, Evidence of chaotic pattern in

solar flux through a reproducible sequence of

period-doubling-type bifurcations, Proceedinqs of Fliqht

Mechanics/Estimation Theory Symposium, May 1991

Ashrafi, S., Goddard Space Flight Center, Flight Dynamics

Division, 554-FDD-91/l12, Future Missions S_udies on Chaotic

Solar Flux (Structural Stability, Attractor Dimension,

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Ashrafi, S., and L. Roszman, Limits on the Predictability of

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Ashrafi, S., Goddard Space Flight Center, Flight Dynamics

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Forecastinq Solar Flux Directly From its Tim_ Seri_s, July

1991

Ashrafi, S., Goddard Space Flight Center, Flight Dynamics

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Unique Canonical Transformation To Transform Dynamo

Equations TO Lorenz Eauations, unpublished manuscript (a)

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SunsPOtS on E-$olitons of Multilevel Turbulence, unpublished

manuscript (b)

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R-I

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_J

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Weiss, N. O., Is the solar cycle an example of deterministicchaos?, in Secular Solar and Geomaqneti¢ Variations in the

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DISTRIBUTION LIST

GSFC

J. Jackson

T. Stengle

J. Teles

FDD CMO

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